Sommerfeld parameter

The Sommerfeld parameter $η$, named after Arnold Sommerfeld, is a dimensionless quantity used in nuclear astrophysics in the calculation of reaction rates between two nuclei and also appears in the definition of the astrophysical S-factor. It is defined as


 * $$ \eta = \frac{Z_1 Z_2 e^2}{4 \pi \epsilon_0 \hbar v} = \alpha Z_1 Z_2 \sqrt{\frac{\mu c^2}{2E}}$$,

where $e$ is the elementary charge, $Z_{1}$ and $Z_{2}$ are the atomic numbers of two interacting nuclides, $v$ is the magnitude of the relative incident velocity in the center-of-mass frame, $α$ is the unitless fine-structure constant, $c$ is the speed of light, and $μ$ is the reduced mass of the two nuclides of interest.

One of its best-known applications is in the exponent of the Gamow factor $P$ (also known as the penetrability factor),


 * $$ P = \exp(-2 \pi \eta) $$,

which is the probability of an s-wave nuclide to penetrate the Coulomb barrier, according to the WKB approximation. This factor is particularly helpful in characterizing the nuclear contribution to low-energy nucleon-scattering cross-sections - namely, through the astrophysical S-factor.

One of the first articles in which the Sommerfeld parameter appeared was published in 1967.